# nLab multiplicative conjunction

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## Idea

In linear logic/linear type theory, one of the two linear versions of logical conjunction is called the multiplicative conjunction and usually denoted “$\otimes$”. The analogous connective in relevance logic is sometimes called the intensional conjunction or the fusion or cotenability, and sometimes denoted “$\circ$”.

The categorical semantics of the multiplicative conjunction is as the tensor product with respect to a (symmetric) monoidal category structure on the collection of types.

There are various fragments of linear logic that contain the multiplicative conjunction. For instance, multiplicative linear logic (MLL) contains $\otimes$ along with its unit $\mathbf{1}$ and also the multiplicative disjunction $\parr$ and its unit $\bot$ and the linear negation $(-)^\bot$. While multiplicative intuitionistic linear logic (MILL) contains $\otimes$ and $\mathbf{1}$ along with the linear implication $\multimap$.

basic symbols used in logic

$\phantom{A}$symbol$\phantom{A}$$\phantom{A}$meaning$\phantom{A}$
$\phantom{A}$$\in$$\phantom{A}$element relation
$\phantom{A}$$\,:$$\phantom{A}$typing relation
$\phantom{A}$$=$$\phantom{A}$equality
$\phantom{A}$$\vdash$$\phantom{A}$$\phantom{A}$entailment / sequent$\phantom{A}$
$\phantom{A}$$\top$$\phantom{A}$$\phantom{A}$true / top$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$$\phantom{A}$false / bottom$\phantom{A}$
$\phantom{A}$$\Rightarrow$$\phantom{A}$implication
$\phantom{A}$$\Leftrightarrow$$\phantom{A}$logical equivalence
$\phantom{A}$$\not$$\phantom{A}$negation
$\phantom{A}$$\neq$$\phantom{A}$negation of equality / apartness$\phantom{A}$
$\phantom{A}$$\notin$$\phantom{A}$negation of element relation $\phantom{A}$
$\phantom{A}$$\not \not$$\phantom{A}$negation of negation$\phantom{A}$
$\phantom{A}$$\exists$$\phantom{A}$existential quantification$\phantom{A}$
$\phantom{A}$$\forall$$\phantom{A}$universal quantification$\phantom{A}$
$\phantom{A}$$\wedge$$\phantom{A}$logical conjunction
$\phantom{A}$$\vee$$\phantom{A}$logical disjunction
$\phantom{A}$$\otimes$$\phantom{A}$$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$$\phantom{A}$multiplicative disjunction$\phantom{A}$

Last revised on November 12, 2019 at 09:19:00. See the history of this page for a list of all contributions to it.